What makes you think that we can visualize higher cubes and spheres? For $n=4$ you might play games like using some sort of time slider to draw the intersection of your object with the $xyz$-hyperplane, but for $n>4$ those kind of hacks will become unavailable very quickly.

The *intuition* behind facts like the ones you quote, isn’t intuition but *computation*. In some sense mathematics builds around our intuition for 2, 3 or maybe even 4-dimensional space, by which I mean that most definitions mimick something in this low dimensional worlds. Yet the definitions are vastly more general in the way that the dimension is inessential, so we might as well try to find out what they do in higher dimensions (thinking of manifolds). It is a pity for sure that we can’t see what is happening there, because sure enough things start to break down. Manifolds become unsmoothable or have multiple distinct smooth structures, classification results are impossible to obtain and spheres become pointy and *computationally* start to look and behave rather alien. To state one example: The *Poincare*-conjecture was one of the millenium problems (ie was on the same level of difficulty than the *Riemann hypothesis* or *$P$ vs $NP$*) and was about $3$-spheres. Higher geometry is **hard**.

On the other hand this is the whole fun about abstract mathematics. Intuitive definitions derived from a small collection of examples soon enough turn out to have more exotic but interesting instances, which makes the definition even more interesting and worthy of studying.